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Functional Semantics for Pregroup Grammars
Gabriel Gaudreault
Concordia University, Montreal
CoCoNat 2015
19 July 2015
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
Goals
Goal:Figuring out a way to do semantics with pregroup grammars thatis intuitive and does not require learning higher-level mathematics
What I am presenting:- A functional calculus that is sensible to the incoming direction ofthe inputs and which has functional composition as main reductionoperation- The 1-1 relation between it and a subset of pregroup grammarsthat is relevant for linguistic analysis- How the system can be used for linguistic analysis
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
Overview
Categorial Grammars
Formal Semantics
Pregroup Grammars
λ-semantics for Pregroup Grammars?
λ↔-calculus
“Curry-Howard” Correspondence
Syntactic & Semantic Analysis
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
Categorial Grammars
Main idea: We can assign mathematical types to words and thencheck whether sentences are grammatical by looking at the stringof their corresponding types and using our derivation rules.
Types s, t := N,S , ... | s/t | s \ t
Reduction rules inspired by arithmetic
Butterflies like orangesN (N \ S) / N N → S
(Noun phrase) (subject \ verb / object) (Noun phrase)
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
Semantics for the Lambek Calculus
Semantics through λ-calculus
λx .λy .love(x , y)
Elimination Rule ↔ Function Application
a : A λx .b(x) : A \ B
b(a) : B
Introduction Rule ↔ Function Abstraction
x : A, Γ ` b : B
Γ ` λx .b : A \ B
In parallel to grammaticality check we do meaning extraction
Alan : NP λw .work(x) : NP \ S
work(Alan) : S
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
Pregroups
(P,→, r , l , ·, 1) : Partially ordered monoid over P, in which everyelement a ∈ P has a right and left adjoint, ar ∈ P, al ∈ Prespectively, subject to
a · ar → 1→ ar · a al · a→ 1→ a · al
(think of arithmetic: 2 ∗ 2−1 = 1)
Has properties such as:
a→ b ⇔ bl → al ⇔ br → ar
arl = alr = a
(a1...an)l = aln...al1
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
Pregroup Grammars
In PG, words get assigned pregroup types π, s, n, n, i , etc. whichcorrespond to syntactic categories
Ordered structure: we can now define relations such as N → π3,s2 → s
The role that inverses (A \ B) and (A / B) played in the originalsyntactic calculus is played by adjoints arb and abl ...
πr s i l i i r i i l i o l nnnln
s
will dance to save humanitymanA
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
Pregroup Grammars
Difference with the Syntactic Calculus:Those new types are associative — (ab)c = a(bc) — and cancombine in much more flexible ways because they are nowconsidered as a list of independent information pieces
abl · bc l → ac l
a/b · b/c 6→ a/c
(in one step)
We can’t really use the λ-calculus to do semantics anymore, as itis not clear what kind of functions our types represent:
(a \ b)/c ⇒ c → a→ b but arbc l ⇒?
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
λ↔-calculus
Termσ Formation Rules
x ∈ Varσx ∈ Termσ
c ∈ Conσc ∈ Termσ
t ∈ TermΩrφ s ∈ Termφrπ
(t)s ∈ TermΩrπ
t ∈ Termπφl s ∈ TermφΩl
t(s) ∈ TermπΩl
t ∈ Termσ x ∈ Varφ
t.x〉 ∈ Termσφl
x ∈ Varφ t ∈ Termσ
〈x .t ∈ Termφrσ
Example:
Alan ∈ Terme 〈x .(x)work ∈ Termerp
(Alan)work ∈ Termp
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
λ↔-calculus - Function Application
Γ1 ` t ∈ TermΩrφ Γ2 ` s ∈ Termφrπ
Γ1, Γ2 ` (t)s ∈ TermΩrπ
Γ1 ` t ∈ Termπφl Γ2 ` s ∈ TermφΩl
Γ1, Γ2 ` t(s) ∈ TermπΩl
Recall in simply typed λ-calculus:
Γ1 ` a : A Γ2 ` λx .f (x) : A→ B
Γ1, Γ2 ` f [a/x ]
Example:
Paul ∈ Terme 〈x .(x)run ∈ Termerp
(Paul)run ∈ Termp
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
λ↔-calculus - Function Abstraction
Γ, x ∈ Varφ ` t ∈ Termσ
Γ ` t.x〉 ∈ Termσφl
x ∈ Varφ, Γ ` t ∈ Termσ
Γ ` 〈x .t ∈ Termφrσ
Recall lambda abstraction:
Γ, x : A ` b : B
Γ ` λx .b(x) : A→ B
Pregroup expansion rule:
x ∈ Vara ` x ∈ Vara
` 〈x .x ∈ Termara
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
Reformulation of Pregroup Types
We redefine pregroup types as a deductive system.
initA ` A
Γ,A ` Bl I
Γ ` BAl
Γ1 ` AB l Γ2 ` BΣllE
Γ1, Γ2 ` AΣl
A, Γ ` Br I
Γ ` ArB
Γ1 ` ΣrA Γ2 ` ArB rEΓ1, Γ2 ` ΣrB
We change the way of looking at ArBC l : it now behaves more likean non-commutative linear functional type A ( B
(
C rather thana cartesian product or sum A⊥ ⊕ B ⊕ C⊥
Σl and Σr stand for a sequences of left and right adjoint types C l1...C
ln,
C r1 ...C
rn respectively.
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
Pregroup−λ↔ Grammars
Easy correspondence between the functional calculus and thosenew pregroup types:
initx : A ` x : A
Γ, x : A ` b : Bl I
Γ ` b.x〉 : BAl
Γ1 ` a : AB l Γ2 ` b : BΣllE
Γ1, Γ2 ` a(b) : AΣl
x : A, Γ ` b : Br I
Γ ` 〈x .b : ArB
Γ1 ` a : ΣrA Γ2 ` b : ArB rEΓ1, Γ2 ` (a)b : ΣrB
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
β-reduction
One of the major differences between that system and somethinglike a λ-calculus with 2 “directional”-λ’s is the β-reduction rules:
t|x〉 (b|x1〉...|xn〉) =β (t)[x := b] |x1〉...|xn〉
when t|x〉 ∈ Termσφl and b|x1〉...|xn〉 ∈ Termφπl1...π
ln
This is the main reason why an untyped λ↔-calculus is not possible
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
β-reduction
t|x〉 (b|x1〉...|xn〉) =β (t)[x := b] |x1〉...|xn〉
when t|x〉 ∈ Termσφl and b|x1〉...|xn〉 ∈ Termφπl1...π
ln
E.g.
the
ı(x).x〉 : nnl
green
green(y).y〉 : nnl
ı(green(y)).y〉 : nnl
Compare this to
theλx .ı(x) : NP/N
green
λy .green(y) : N/N z : N
green(z) : N
ı(green(z)) : NP
λz .ı(green(z)) : NP/N
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
β-reduction
t|x〉 (b|x1〉...|xn〉) =β (t)[x := b] |x1〉...|xn〉
when t|x〉 ∈ Termσφl and b|x1〉...|xn〉 ∈ Termφπl1...π
ln
We still need to be able to pass abstracted predicates though
Someone
∃x .(x)y .y〉 : s(πr3s)lruns
〈z .(z)run : πr3s
∃x .(x)〈z .(z)run : s
∃x .(x)run : s
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
New Typing Structure
The pregroup types we will end up using in our analyses only forma subset of the ones possible in the algebraic formulation. Forinstance, there are
No more types of the form Ar , ArrB, AlB, AB, etc.
No more relations such as (AB)l ↔ B lAl
Only one kind of contraction is possible: between“disconnected” types
It is totally fine: we will end up not needing those relations andtypes at all, and our contraction rule is sufficient
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
New Typing Structure
Predictions:
give never needs its equivalent type i(op)l
give a star to Bobiplo l nnl n pnl N
→ iplo l n pnl n
→ iplo l o p
→ ipl p → i
for does not need φ(oj)l either
John wants for Mary to liveN πr3sφl φj lo l N ji l i
→ π3 πr3sφl φj lo l o j
→ sφl φj l j
→ sφl φ → s
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
New Typing Structure
More examples:
somebody
ss lπ?πr
Edward likes pies
s
JohnN
who
nr ns lπlCarlπ
I like ice creams
n
JohnN
who
nr n(πs)l?πs
n
JohnN
whom
nr no l ls lI like ice cream
s
?
o l
n
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
Flexibility of Derivations
the
ı(x)|x〉 : nnl
green
green(y)|y〉 : nnllE
ı(green(y))|y〉 : nnl
apple
apple : nlE
ı(green(apple)) : n
the
ı(x)|x〉 : nnl
green
green(y)|y〉 : nnl
apple
apple : nlE
green(apple) : nlE
ı(green(apple)) : n
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
Constituency Analysis
fliesplural(fly(x)|x〉) : n
in
〈y |(y)in(x)|x〉 : nrnnl
the
ı(x)|x〉 : nnl
sky
sky : nlE
ı(sky) : nlE〈y |(y)in(ı(sky)) : nrn
lE(plural(fly(x)|x〉)in(ı(sky)) : n
flies〈z |fly(z) : πr s
in
〈y |(y)in(x)|x〉 : sr snllE〈z |(fly(z))in(x)|x〉 : πr snl
the
ı(x)|x〉 : nnl
sky
sky : nlE
ı(sky) : nlE〈z |(fly(z))in(ı(sky)) : πr s
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
Examples
Sentence with Quantifiers
First reading: Everybody has a special someone, who might all bedifferent from eachother
everybody
∀((person(z) → (z)x|z〉))|x〉 : s(πr3s)
l
loves
〈x|(x)love(y)|y〉 : πr3so
l
somebody
〈z|∃((person(y) → z(y))|y〉) : (so l )r s
〈x|∃((person(y) → (x)love(y))|y〉) : πr3s
∀((person(z) → ∃((person(y) → (z)love(y))|y〉))|z〉) : s
Second reading: There’s someone who everyone’s in love with
everybody
∀((person(z) → (z)x|z〉))|x〉 : s(πr3s)
l
loves
〈x|(x)love(y)|y〉 : πr3so
l
∀((person(z) → (z)love(y)|z〉))|y〉 : so l
somebody
〈z|∃((person(x) → z(x))|x〉) : (so l )r s
∃((person(x) → ∀((person(z) → (z)love(x)|z〉)))|x〉) : s
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
Future
More in-depth analysis of semantic power of the new calculus,i.e. what does an associative & compositional semantic layerimply for semantic analysis?
Proving Church-Rosser Property
Category theoretical analysis of the new calculus
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars
The End
Thank you for listening
Special thanks to: Alan Bale, Claudia Casadio, and Robert Seelyfor their help in this project
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars